# Seminars (GCSW01)

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Event When Speaker Title Presentation Material
GCSW01 8th July 2019
10:00 to 11:00
Hans Munthe-Kaas Why B-series, rooted trees, and free algebras? - 1
"We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results”. This quote by Alain Connes summarises nicely the mathematical depth and scope of the theory of Butcher's B-series. The aim of this joined lecture is to answer the question posed in the title by drawing a line from B-series to those far-reaching conceptional results they originated. Unfolding the precise mathematical picture underlying B-series requires a combination of different perspectives and tools from geometry (connections); analysis (generalisations of Taylor expansions), algebra (pre-/post-Lie and Hopf algebras) and combinatorics (free algebras on rooted trees). This summarises also the scope of these lectures.   In the first lecture we will outline the geometric foundations of B-series, and their cousins Lie-Butcher series. The latter is adapted to studying differential equations on manifolds. The theory of connections and parallel transport will be explained. In the second and third lectures we discuss the algebraic and combinatorial structures arising from the study of invariant connections. Rooted trees play a particular role here as they provide optimal index sets for the terms in Taylor series and generalisations thereof. The final lecture will discuss various applications of the theory in the numerical analysis of integration schemes.
GCSW01 8th July 2019
11:30 to 12:30
Charlie Elliott PDEs in Complex and Evolving Domains I
GCSW01 8th July 2019
14:00 to 15:00
Elizabeth Mansfield Introduction to Lie groups and algebras - 1
In this series of three lectures, I will give a gentle introduction to Lie groups and their associated Lie algebras, concentrating on the major examples of importance in applications. I will next discuss Lie group actions, their invariants, their associated infinitesimal vector fields, and a little on moving frames. The final topic will be Noether’s Theorem, which yields conservations laws for variational problems which are invariant under a Lie group action. Time permitting, I will show a little of the finite difference and finite element versions of Noether’s Theorem.

In this first talk, I will consider some of the simplest and most useful Lie groups. I will show how to derive their Lie algebras, and will further discuss the Adjoint action of the group on its Lie algebra, the Lie bracket on the algebra, and the exponential map.
GCSW01 8th July 2019
15:00 to 16:00
Christopher Budd Modified error estimates for discrete variational derivative methods
My three talks will be an exploration of
geometric integration methods in the context

of the numerical solution of PDEs. I will look
both at discrete variational methods

and their analysis using modified equations, and
also of the role of adaptivity in

studying and retaining qualitative features of
PDEs.
GCSW01 8th July 2019
16:30 to 17:30
Paola Francesca Antonietti High-order Discontinuous Galerkin methods for the numerical modelling of earthquake ground motion
A number of challenging geophysical applications requires a flexible representation of the geometry and an accurate approximation of the solution field. Paradigmatic examples include seismic wave propagation and fractured reservoir simulations. The main challenges are i) the complexity of the physical domain, due to the presence of localized geological irregularities, alluvial basins, faults and fractures; ii) the heterogeneities in the medium, with significant and sharp contrasts; and iii) the coexistence of different physical models. The high-order discontinuous Galerkin FEM possesses the built-in flexibility to naturally accommodate both non-matching meshes, possibly made of polygonal and polyhedral elements, and high-order approximations in any space dimension. In this talk I will discuss recent advances in the development and analysis of high-order DG methods for the numerical approximation of seismic wave propagation phenomena. I will analyse the stability and the theoretical properties of the scheme and present some simulations of real large-scale seismic events in three-dimensional complex media.
GCSW01 9th July 2019
09:00 to 10:00
Douglas Arnold Finite Element Exterior Calculus - 1
These lectures aim to provide an introduction and overview of Finite Element Exterior Calculus, a transformative approach to designing and understanding numerical methods for partial differential equations. The first lecture will introduce some of the key tools--chain complexes and their cohomology, closed operators in Hilbert space, and their marriage in the notion of Hilbert complexes--and explore their application to PDEs. The lectures will continue with a study of the properties needed to effectively discretize Hilbert complexes, illustrating the abstract framework on the concrete example of the de Rham complex and its applications to problems such as Maxwell's equation. The third lecture will get into differential forms and their discretization by finite elements, bringing in new tools like the Koszul complex and bounded cochain projections and revealing the Periodic Table of Finite Elements. Finally in the final lecture we will examine new complexes, their discretization, and applications.
GCSW01 9th July 2019
10:00 to 11:00
Elizabeth Mansfield Introduction to Lie groups and algebras - 2
I will discuss Lie group actions, their invariants, their associated infinitesimal vector fields, and a little on moving frames.
GCSW01 9th July 2019
11:30 to 12:30
Kurusch Ebrahimi-Fard Why B-series, rooted trees, and free algebras? - 2
"We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results”. This quote by Alain Connes summarises nicely the mathematical depth and scope of the theory of Butcher's B-series. The aim of this joined lecture is to answer the question posed in the title by drawing a line from B-series to those far-reaching conceptional results they originated. Unfolding the precise mathematical picture underlying B-series requires a combination of different perspectives and tools from geometry (connections); analysis (generalisations of Taylor expansions), algebra (pre-/post-Lie and Hopf algebras) and combinatorics (free algebras on rooted trees). This summarises also the scope of these lectures.   In the first lecture we will outline the geometric foundations of B-series, and their cousins Lie-Butcher series. The latter is adapted to studying differential equations on manifolds. The theory of connections and parallel transport will be explained. In the second and third lectures we discuss the algebraic and combinatorial structures arising from the study of invariant connections. Rooted trees play a particular role here as they provide optimal index sets for the terms in Taylor series and generalisations thereof. The final lecture will discuss various applications of the theory in the numerical analysis of integration schemes.
GCSW01 9th July 2019
14:00 to 15:00
Charlie Elliott PDEs in Complex and Evolving Domains II
GCSW01 9th July 2019
15:00 to 16:00
Christian Lubich Variational Gaussian wave packets revisited
The talk reviews Gaussian wave packets that evolve according to the Dirac-Frenkel time-dependent variational principle for the semi-classically scaled Schr\"odinger equation. Old and new results on the approximation to the wave function are given, in particular an $L^2$ error bound that goes back to Hagedorn (1980) in a non-variational setting, and a new error bound for averages of observables with a Weyl symbol, which shows the double approximation order in the semi-classical scaling parameter in comparison with the norm estimate.

The variational equations of motion in Hagedorn's parametrization of the Gaussian are presented. They show a perfect quantum-classical correspondence and allow us to read off directly that the Ehrenfest time is determined by the Lyapunov exponent of the classical equations of motion.

A variational splitting integrator is formulated and its remarkable conservation and approximation properties are discussed. A new result shows that the integrator approximates averages of observables with the full order in the time stepsize, with an error constant that is uniform in the semiclassical parameter.

The material presented here for variational Gaussians is part of an Acta Numerica review article on computational methods for quantum dynamics in the semi-classical regime, which is currently in preparation in joint work with Caroline Lasser.

GCSW01 10th July 2019
09:00 to 10:00
Christopher Budd Blow-up in PDES and how to compute it
My three talks will be an exploration of
geometric integration methods in the context

of the numerical solution of PDEs. I will look
both at discrete variational methods

and their analysis using modified equations, and
also of the role of adaptivity in

studying and retaining qualitative features of
PDEs.
GCSW01 10th July 2019
10:00 to 11:00
Douglas Arnold Finite Element Exterior Calculus - 2
These lectures aim to provide an introduction and overview of Finite Element Exterior Calculus, a transformative approach to designing and understanding numerical methods for partial differential equations. The first lecture will introduce some of the key tools--chain complexes and their cohomology, closed operators in Hilbert space, and their marriage in the notion of Hilbert complexes--and explore their application to PDEs. The lectures will continue with a study of the properties needed to effectively discretize Hilbert complexes, illustrating the abstract framework on the concrete example of the de Rham complex and its applications to problems such as Maxwell's equation. The third lecture will get into differential forms and their discretization by finite elements, bringing in new tools like the Koszul complex and bounded cochain projections and revealing the Periodic Table of Finite Elements. Finally in the final lecture we will examine new complexes, their discretization, and applications.
GCSW01 10th July 2019
11:30 to 12:30
Charlie Elliott PDEs in Complex and Evolving Domains III
GCSW01 11th July 2019
09:00 to 10:00
Charlie Elliott PDEs in Complex and Evolving Domains IV
GCSW01 11th July 2019
10:00 to 11:00
Beth Wingate An introduction to time-parallel methods
I will give an introduction to time-parallel methods and discuss how they apply to PDEs, in particular those with where resonance plays an important role. I’ll give some examples from ODEs before finally discussing PDEs.
GCSW01 11th July 2019
11:30 to 12:30
Douglas Arnold Finite Element Exterior Calculus - 3
These lectures aim to provide an introduction and overview of Finite Element Exterior Calculus, a transformative approach to designing and understanding numerical methods for partial differential equations. The first lecture will introduce some of the key tools--chain complexes and their cohomology, closed operators in Hilbert space, and their marriage in the notion of Hilbert complexes--and explore their application to PDEs. The lectures will continue with a study of the properties needed to effectively discretize Hilbert complexes, illustrating the abstract framework on the concrete example of the de Rham complex and its applications to problems such as Maxwell's equation. The third lecture will get into differential forms and their discretization by finite elements, bringing in new tools like the Koszul complex and bounded cochain projections and revealing the Periodic Table of Finite Elements. Finally in the final lecture we will examine new complexes, their discretization, and applications.
GCSW01 11th July 2019
14:00 to 15:00
Kurusch Ebrahimi-Fard Why B-series, rooted trees, and free algebras? - 3
"We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results”. This quote by Alain Connes summarises nicely the mathematical depth and scope of the theory of Butcher's B-series. The aim of this joined lecture is to answer the question posed in the title by drawing a line from B-series to those far-reaching conceptional results they originated. Unfolding the precise mathematical picture underlying B-series requires a combination of different perspectives and tools from geometry (connections); analysis (generalisations of Taylor expansions), algebra (pre-/post-Lie and Hopf algebras) and combinatorics (free algebras on rooted trees). This summarises also the scope of these lectures.   In the first lecture we will outline the geometric foundations of B-series, and their cousins Lie-Butcher series. The latter is adapted to studying differential equations on manifolds. The theory of connections and parallel transport will be explained. In the second and third lectures we discuss the algebraic and combinatorial structures arising from the study of invariant connections. Rooted trees play a particular role here as they provide optimal index sets for the terms in Taylor series and generalisations thereof. The final lecture will discuss various applications of the theory in the numerical analysis of integration schemes.
GCSW01 11th July 2019
15:00 to 16:00
Elizabeth Mansfield Noether's conservation laws - smooth and discrete
The final topic will be Noether’s Theorem, which yields conservation laws for variational problems which are invariant under a Lie group action. . I will show a little of the finite difference and finite element versions of Noether’s Theorem.
GCSW01 11th July 2019
16:30 to 17:30
Panel
GCSW01 12th July 2019
09:00 to 10:00
Reinout Quispel Discrete Darboux polynomials and the preservation of measure and integrals of ordinary differential equations
Preservation of phase space volume (or more generally measure), first integrals (such as energy), and second integrals have been important topics in geometric numerical integration for more than a decade, and methods have been developed to preserve each of these properties separately. Preserving two or more geometric properties simultaneously, however, has often been difficult, if not impossible. Then it was discovered that Kahan’s ‘unconventional’ method seems to perform well in many cases [1]. Kahan himself, however, wrote: “I have used these unconventional methods for 24 years without quite understanding why they work so well as they do, when they work.” The first approximation to such an understanding in computational terms was: Kahan’s method works so well because
1.         It is very successful at preserving multiple quantities simultaneously, eg modified energy and modified measure.
2.         It is linearly implicit
3.         It is the restriction of a Runge-Kutta method
However, point 1 above raises a further obvious question: Why does Kahan’s method preserve both certain (modified) first integrals and certain (modified) measures? In this talk we invoke Darboux polynomials to try and answer this question. The method of Darboux polynomials (DPs) for ODEs was introduced by Darboux to detect rational integrals. Very recently we have advocated the use of DPs for discrete systems [2,3]. DPs provide a unified theory for the preservation of polynomial measures and second integrals, as well as rational first integrals. In this new perspective the answer we propose to the above question is: Kahan’s method works so well because it is good at preserving (modified) Darboux polynomials. If time permits we may discuss extensions to polarization methods.

[1] Petrera et al, Regular and Chaotic Dynamics 16 (2011), 245–289.
[2] Celledoni et al, arxiv:1902.04685.
[3] Celledoni et al, arxiv:1902.04715.
GCSW01 12th July 2019
10:00 to 10:30
James Jackaman Lie symmetry preserving finite element methods
Through the mathematical construction of finite element methods, the “standard” finite element method will often preserve the underlying symmetry of a given differential equation. However, this is not always the case, and while historically much attention has been paid to the preserving of conserved quantities the preservation of Lie symmetries is an open problem. We introduce a methodology for the design of arbitrary order finite element methods which preserve the underlying Lie symmetries through an equivariant moving frames based invariantization procedure.
GCSW01 12th July 2019
10:30 to 11:00
Candan Güdücü Port-Hamiltonian Systems
The framework of port-Hamiltonian systems (PH systems) combines both the Hamiltonian approach and the network approach, by associating with the interconnection structure of the network model a geometric structure given by a Dirac structure. In this talk, I introduce port-Hamiltonian (pH) systems and their underlying Dirac structures. Then, a coordinate-based representation of PH systems and some properties are shown. A Lanczos method for the solution of linear systems with nonsymmetric coefficient matrices and its application to pH systems are presenred.
GCSW01 12th July 2019
11:30 to 12:30
Christopher Budd Adaptivity and optimal transport
My three talks will be an exploration of
geometric integration methods in the context

of the numerical solution of PDEs. I will look
both at discrete variational methods

and their analysis using modified equations, and
also of the role of adaptivity in

studying and retaining qualitative features of
PDEs.
GCSW01 12th July 2019
14:00 to 15:00
Douglas Arnold Finite element exterior calculus - 4
These lectures aim to provide an introduction and overview of Finite Element Exterior Calculus, a transformative approach to designing and understanding numerical methods for partial differential equations. The first lecture will introduce some of the key tools--chain complexes and their cohomology, closed operators in Hilbert space, and their marriage in the notion of Hilbert complexes--and explore their application to PDEs. The lectures will continue with a study of the properties needed to effectively discretize Hilbert complexes, illustrating the abstract framework on the concrete example of the de Rham complex and its applications to problems such as Maxwell's equation. The third lecture will get into differential forms and their discretization by finite elements, bringing in new tools like the Koszul complex and bounded cochain projections and revealing the Periodic Table of Finite Elements. Finally in the final lecture we will examine new complexes, their discretization, and applications.
GCSW01 12th July 2019
15:00 to 16:00
Hans Munthe-Kaas Why B-series, rooted trees, and free algebras? - 2
"We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results”. This quote by Alain Connes summarises nicely the mathematical depth and scope of the theory of Butcher's B-series. The aim of this joined lecture is to answer the question posed in the title by drawing a line from B-series to those far-reaching conceptional results they originated. Unfolding the precise mathematical picture underlying B-series requires a combination of different perspectives and tools from geometry (connections); analysis (generalisations of Taylor expansions), algebra (pre-/post-Lie and Hopf algebras) and combinatorics (free algebras on rooted trees). This summarises also the scope of these lectures.   In the first lecture we will outline the geometric foundations of B-series, and their cousins Lie-Butcher series. The latter is adapted to studying differential equations on manifolds. The theory of connections and parallel transport will be explained. In the second and third lectures we discuss the algebraic and combinatorial structures arising from the study of invariant connections. Rooted trees play a particular role here as they provide optimal index sets for the terms in Taylor series and generalisations thereof. The final lecture will discuss various applications of the theory in the numerical analysis of integration schemes.